Principles of Virtual work

Introduction

Virtual work is a fundamental concept in structural analysis that allows engineers to determine displacements and internal forces in deformable bodies without solving complex equations. By understanding the principles of virtual work, engineers can analyze complex structures and make informed design decisions. This article provides an overview of the key concepts and principles associated with virtual work, as well as step-by-step walkthroughs of typical problems and real-world applications.

Importance of Virtual Work

Virtual work plays a crucial role in structural analysis as it provides a systematic approach to analyze complex structures. It allows engineers to determine displacements and internal forces without solving complex equations, making the analysis process more efficient and accurate.

Fundamentals of Virtual Work

Virtual work is based on the principle of equilibrium, which states that the sum of the external forces acting on a body is equal to zero. This principle is applied to deformable bodies, which are bodies that undergo deformation under external loads.

Definition of Virtual Work

Virtual work can be defined as the work done by virtual displacements in a structure under the action of external forces. Virtual displacements are hypothetical displacements that satisfy the equilibrium conditions of the structure. By considering the work done by these virtual displacements, engineers can determine the displacements and internal forces in the structure.

Key Concepts and Principles

Virtual Work Principle

The virtual work principle states that the work done by the external forces acting on a structure is equal to the work done by the internal forces resulting from the deformations caused by these external forces. This principle is based on the principle of conservation of energy and is derived from the principle of equilibrium.

The virtual work principle can be mathematically expressed as:

$$\delta W_{ext} = \delta W_{int}$$

where:

• $$\delta W_{ext}$$ is the work done by the external forces
• $$\delta W_{int}$$ is the work done by the internal forces

By applying this principle, engineers can determine the displacements and internal forces in a structure.

Application of Virtual Work Principle

The virtual work principle is applied by considering virtual displacements in the structure. Virtual displacements are hypothetical displacements that satisfy the equilibrium conditions of the structure. By considering the work done by these virtual displacements, engineers can determine the displacements and internal forces in the structure.

To apply the virtual work principle, the following steps are typically followed:

1. Define the external forces acting on the structure.
2. Assume virtual displacements that satisfy the equilibrium conditions of the structure.
3. Calculate the work done by the external forces and the work done by the internal forces resulting from the virtual displacements.
4. Equate the work done by the external forces to the work done by the internal forces to obtain the desired displacements and internal forces.

Importance of Virtual Work Principle

The virtual work principle is of great importance in structural analysis as it allows engineers to determine displacements and internal forces in structures without solving complex equations. This principle provides a systematic approach to analyze complex structures and is particularly useful in determining the behavior of deformable bodies.

Deformable Bodies

Deformable bodies are bodies that undergo deformation under external loads. These bodies can be subjected to various types of loads, such as axial loads, shear forces, and bending moments. The behavior of deformable bodies is characterized by the relationship between deformations and internal forces.

Relationship between Deformations and Internal Forces

The relationship between deformations and internal forces in deformable bodies is described by the constitutive equations. These equations relate the deformations, such as displacements and strains, to the internal forces, such as stresses and moments.

By considering the relationship between deformations and internal forces, engineers can determine the internal forces resulting from given deformations or vice versa.

Importance of Considering Deformable Bodies

It is important to consider deformable bodies in structural analysis as they accurately represent the behavior of real structures. By considering the deformations and internal forces in deformable bodies, engineers can make informed design decisions and ensure the structural integrity and safety of the structure.

Step-by-Step Walkthrough of Typical Problems and Solutions

This section provides step-by-step walkthroughs of typical problems that can be solved using the virtual work principle.

Problem 1: Determining Displacements Using Virtual Work Principle

Problem Scenario

Consider a simply supported beam subjected to a concentrated load at the midpoint. Determine the vertical displacement at the midpoint of the beam.

Given:

• Length of the beam: $$L$$
• Concentrated load: $$P$$

Solution

Step 1: Define the external forces

The external force acting on the beam is the concentrated load $$P$$.

Step 2: Assume virtual displacements

Assume a virtual displacement at the midpoint of the beam.

Step 3: Calculate the work done by the external forces

The work done by the external force $$P$$ can be calculated as:

$$\delta W_{ext} = P \cdot \delta y$$

where $$\delta y$$ is the virtual displacement.

Step 4: Calculate the work done by the internal forces

The work done by the internal forces can be calculated as:

$$\delta W_{int} = \frac{1}{2} \cdot k \cdot \delta y^2$$

where $$k$$ is the stiffness of the beam.

Step 5: Equate the work done by the external forces to the work done by the internal forces

Equate $$\delta W_{ext}$$ to $$\delta W_{int}$$ and solve for $$\delta y$$ to obtain the vertical displacement at the midpoint of the beam.

Problem 2: Determining Internal Forces Using Virtual Work Principle

Problem Scenario

Consider a cantilever beam subjected to a uniformly distributed load. Determine the bending moment at a specific location along the beam.

Given:

• Length of the beam: $$L$$
• Uniformly distributed load: $$w$$
• Location of interest: $$x$$

Solution

Step 1: Define the external forces

The external force acting on the beam is the uniformly distributed load $$w$$.

Step 2: Assume virtual displacements

Assume a virtual displacement at the location of interest.

Step 3: Calculate the work done by the external forces

The work done by the external force $$w$$ can be calculated as:

$$\delta W_{ext} = w \cdot \delta y$$

where $$\delta y$$ is the virtual displacement.

Step 4: Calculate the work done by the internal forces

The work done by the internal forces can be calculated as:

$$\delta W_{int} = M \cdot \delta \theta$$

where $$M$$ is the bending moment at the location of interest and $$\delta \theta$$ is the virtual rotation.

Step 5: Equate the work done by the external forces to the work done by the internal forces

Equate $$\delta W_{ext}$$ to $$\delta W_{int}$$ and solve for $$M$$ to obtain the bending moment at the location of interest.

Real-World Applications and Examples

This section provides real-world applications and examples of the virtual work principle.

Application 1: Determining Displacements and Internal Forces in a Bridge Structure

Bridge Structure

Consider a bridge structure subjected to various loads, such as dead loads, live loads, and temperature changes. The bridge structure consists of beams, columns, and cables.

Loading Conditions

The bridge structure is subjected to dead loads, which include the weight of the bridge components, and live loads, which include the weight of vehicles and pedestrians. In addition, the bridge structure is subjected to temperature changes, which cause thermal expansions and contractions.

Application of Virtual Work Principle

The virtual work principle can be applied to determine the displacements and internal forces in the bridge structure. By considering the work done by the external forces and the work done by the internal forces resulting from the deformations caused by these external forces, engineers can analyze the behavior of the bridge structure.

Application 2: Determining Displacements and Internal Forces in a Building Frame

Building Frame

Consider a building frame subjected to various loads, such as wind loads, seismic loads, and temperature changes. The building frame consists of columns, beams, and slabs.

Loading Conditions

The building frame is subjected to wind loads, which exert horizontal forces on the structure. In addition, the building frame is subjected to seismic loads, which cause vibrations and deformations. Furthermore, the building frame is subjected to temperature changes, which cause thermal expansions and contractions.

Application of Virtual Work Principle

The virtual work principle can be applied to determine the displacements and internal forces in the building frame. By considering the work done by the external forces and the work done by the internal forces resulting from the deformations caused by these external forces, engineers can analyze the behavior of the building frame.

Advantages and Disadvantages of Virtual Work

Advantages

The virtual work principle offers several advantages in structural analysis:

1. Provides a systematic approach to analyze complex structures

The virtual work principle provides a systematic approach to analyze complex structures. By considering the work done by the external forces and the work done by the internal forces resulting from the deformations caused by these external forces, engineers can determine the displacements and internal forces in the structure.

1. Allows for the determination of displacements and internal forces without solving complex equations

The virtual work principle allows engineers to determine displacements and internal forces without solving complex equations. By considering the work done by the external forces and the work done by the internal forces resulting from the deformations caused by these external forces, engineers can obtain the desired results.

Disadvantages

The virtual work principle has some limitations and disadvantages:

1. Requires a good understanding of the virtual work principle and its application

To apply the virtual work principle effectively, engineers need to have a good understanding of the principle and its application. This requires knowledge of structural analysis and mechanics.

1. May not be suitable for all types of structures or loading conditions

The virtual work principle may not be suitable for all types of structures or loading conditions. In some cases, other methods or principles may be more appropriate and accurate.

Conclusion

In conclusion, the principles of virtual work are essential in structural analysis. By understanding the virtual work principle and its application, engineers can determine displacements and internal forces in deformable bodies without solving complex equations. The virtual work principle provides a systematic approach to analyze complex structures and offers advantages such as efficiency and accuracy. However, it requires a good understanding of the principle and may not be suitable for all types of structures or loading conditions. Overall, virtual work is a valuable tool in structural analysis and plays a significant role in ensuring the structural integrity and safety of various structures.

Summary

Virtual work is a fundamental concept in structural analysis that allows engineers to determine displacements and internal forces in deformable bodies without solving complex equations. By understanding the principles of virtual work, engineers can analyze complex structures and make informed design decisions. The virtual work principle states that the work done by the external forces is equal to the work done by the internal forces resulting from the deformations caused by these external forces. Deformable bodies are bodies that undergo deformation under external loads, and their behavior is characterized by the relationship between deformations and internal forces. The virtual work principle provides a systematic approach to analyze complex structures and offers advantages such as efficiency and accuracy. However, it requires a good understanding of the principle and may not be suitable for all types of structures or loading conditions.

Analogy

Imagine a rubber band stretched between two points. When you pull on the rubber band, it stretches and deforms. The amount of stretching and deformation depends on the force applied. In this analogy, the rubber band represents the deformable body, and the force applied represents the external force. By considering the work done by the external force and the work done by the internal forces resulting from the deformation of the rubber band, you can determine the displacements and internal forces in the rubber band.